Best Known (123−18, 123, s)-Nets in Base 4
(123−18, 123, 29130)-Net over F4 — Constructive and digital
Digital (105, 123, 29130)-net over F4, using
- net defined by OOA [i] based on linear OOA(4123, 29130, F4, 18, 18) (dual of [(29130, 18), 524217, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(4123, 262170, F4, 18) (dual of [262170, 262047, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(4123, 262176, F4, 18) (dual of [262176, 262053, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- linear OA(4118, 262144, F4, 18) (dual of [262144, 262026, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(491, 262144, F4, 14) (dual of [262144, 262053, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(45, 32, F4, 3) (dual of [32, 27, 4]-code or 32-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(4123, 262176, F4, 18) (dual of [262176, 262053, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(4123, 262170, F4, 18) (dual of [262170, 262047, 19]-code), using
(123−18, 123, 131088)-Net over F4 — Digital
Digital (105, 123, 131088)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4123, 131088, F4, 2, 18) (dual of [(131088, 2), 262053, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4123, 262176, F4, 18) (dual of [262176, 262053, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- linear OA(4118, 262144, F4, 18) (dual of [262144, 262026, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(491, 262144, F4, 14) (dual of [262144, 262053, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(45, 32, F4, 3) (dual of [32, 27, 4]-code or 32-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- OOA 2-folding [i] based on linear OA(4123, 262176, F4, 18) (dual of [262176, 262053, 19]-code), using
(123−18, 123, large)-Net in Base 4 — Upper bound on s
There is no (105, 123, large)-net in base 4, because
- 16 times m-reduction [i] would yield (105, 107, large)-net in base 4, but